Some examples of nontrivial homotopy groups of modules
The concept of the homotopy theory of modules was discovered by Peter Hilton as a result of his trip in 1955 to Warsaw, Poland, to work with Karol Borsuk, and to Zurich, Switzerland, to work with Beno Eckmann. The idea was to produce an analog of homotopy theory in topology. Yet, unlike homotopy the...
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Wiley
2001-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171201005373 |
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| author | C. Joanna Su |
| author_facet | C. Joanna Su |
| author_sort | C. Joanna Su |
| collection | DOAJ |
| description | The concept of the homotopy theory of modules was discovered by Peter Hilton as a result of his trip in 1955 to Warsaw, Poland, to work with Karol Borsuk, and to Zurich, Switzerland, to work with Beno Eckmann. The idea was to produce an analog of homotopy theory in topology. Yet, unlike homotopy theory in topology, there are two homotopy theories of modules, the injective theory, π¯n(A,B), and the projective theory, π¯n(A,B). They are dual, but not isomorphic.
In this paper, we deliver and carry out the precise calculation of
the first known nontrivial examples of absolute homotopy groups of
modules, namely, π¯n(ℚ/ℤ,ℚ/ℤ), π¯n(ℤ,ℚ/ℤ), and π¯n(ℤ,ℤ), where ℚ/ℤ and ℤ
are regarded as ℤCk-modules with trivial action. One interesting phenomenon of the results is the periodicity
of these homotopy groups, just as for the Ext groups. |
| format | Article |
| id | doaj-art-da07d0423a5e4fbf93de3e7f6cb45005 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2001-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-da07d0423a5e4fbf93de3e7f6cb450052025-02-03T01:02:19ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252001-01-0127318919510.1155/S0161171201005373Some examples of nontrivial homotopy groups of modulesC. Joanna Su0Department of Mathematical Sciences, State University of New York at Binghamton, Binghamton 13902-6000, NY, USAThe concept of the homotopy theory of modules was discovered by Peter Hilton as a result of his trip in 1955 to Warsaw, Poland, to work with Karol Borsuk, and to Zurich, Switzerland, to work with Beno Eckmann. The idea was to produce an analog of homotopy theory in topology. Yet, unlike homotopy theory in topology, there are two homotopy theories of modules, the injective theory, π¯n(A,B), and the projective theory, π¯n(A,B). They are dual, but not isomorphic. In this paper, we deliver and carry out the precise calculation of the first known nontrivial examples of absolute homotopy groups of modules, namely, π¯n(ℚ/ℤ,ℚ/ℤ), π¯n(ℤ,ℚ/ℤ), and π¯n(ℤ,ℤ), where ℚ/ℤ and ℤ are regarded as ℤCk-modules with trivial action. One interesting phenomenon of the results is the periodicity of these homotopy groups, just as for the Ext groups.http://dx.doi.org/10.1155/S0161171201005373 |
| spellingShingle | C. Joanna Su Some examples of nontrivial homotopy groups of modules International Journal of Mathematics and Mathematical Sciences |
| title | Some examples of nontrivial homotopy groups of modules |
| title_full | Some examples of nontrivial homotopy groups of modules |
| title_fullStr | Some examples of nontrivial homotopy groups of modules |
| title_full_unstemmed | Some examples of nontrivial homotopy groups of modules |
| title_short | Some examples of nontrivial homotopy groups of modules |
| title_sort | some examples of nontrivial homotopy groups of modules |
| url | http://dx.doi.org/10.1155/S0161171201005373 |
| work_keys_str_mv | AT cjoannasu someexamplesofnontrivialhomotopygroupsofmodules |